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The image contains three mathematical problems:

1. **Problem 19**:
   Consider five points in the plane, with no three of them collinear. Every pair of points among them is joined by a line. How many ways can we color these lines red or blue so that no three of the points form a triangle with lines of the same color?

2. **Problem 20**:
   On a natural number, you are allowed two operations: (1) multiply by 2, or (2) subtract 3 from it. For example, starting from 11, you can reach 16 in two steps, and you cannot do it in less than two steps. Starting from 11, what is the least number of steps required to reach 172?

3. **Problem 21**:
   An integer \( n \) is such that \( \frac{n}{9} = \frac{n}{12} = \frac{n}{14} \). Is \( n \) a three-digit number with equal digits, and if \( n \) is a four-digit number with the digits 2, 9, 1, 4 in some order, what is the remainder when \( n \) is divided by 100?

Let's solve these problems one by one:

### Problem 19:
This is a question about graph theory, specifically about edge coloring in a complete graph \( K_5 \). The problem is asking for the number of ways to 2-color the edges such that there is no monochromatic triangle. This problem is related to Ramsey theory, and the exact number of such colorings can be complex to determine directly.

### Problem 20:
Starting from 11, find the least number of steps required to reach 172 using the operations:
1. Multiply by 2.
2. Subtract 3.

We need to find the shortest path using these operations. Let's calculate this step-by-step to find the optimal route.

### Problem 21:
To solve \( n \) for \( \frac{n}{9} = \frac{n}{12} = \frac{n}{14} \), we need \( n \) to be a common multiple of 9, 12, and 14. Calculating the least common multiple (LCM) of these numbers will give us \( n \). If \( n \) is a three-digit number with equal digits, it can only be 111, 222, ..., 999. If \( n \) is a four-digit number with the digits 2, 9, 1, 4 in some order, we need to find \( n \mod 100 \).

Would you like a detailed solution for each problem? Here are some related questions:

1. What is the mathematical significance of Ramsey theory in graph coloring problems?
2. How do you calculate the least common multiple (LCM) of multiple numbers?
3. What is the algorithm to find the shortest path using given operations in a problem like Problem 20?
4. How can combinatorial mathematics help solve edge-coloring problems in graphs?
5. What are common strategies for solving Diophantine equations in integer problems?

**Tip:** Breaking down complex problems into smaller, manageable steps often simplifies the process and leads to clearer solutions.

Consider five points in the plane, with no three of them collinear. Every pair of points among them is joined by a line. How many ways can we color these lines red or blue so that no three of the points form a triangle with lines of the same color?
On a natural number, you are allowed two operations: (1) multiply by 2, or (2) subtract 3 from it. For example, starting from 11, what is the least number of steps required to reach 172?
An integer n is such that n/9 = n/12 = n/14. Is n a three-digit number with equal digits, and what is the remainder when a four-digit number with the digits 2, 9, 1, 4 is divided by 100?

Explore challenging mathematical problems involving graph theory, number operations, and least common multiple. Learn how to solve a 5-point edge coloring problem and how to reach 172 from 11 using basic operations. We also explore an integer division problem involving LCMs. Suitable for Grades 9-12.

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